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OCT
7
2003

Modal Logic

A modal is an expression (like ‘necessarily’ or
‘possibly’) that is used to qualify the truth of a
judgement. Modal logic is, strictly speaking, the study of the
deductive behavior of the expressions ‘it is necessary
that’ and ‘it is possible that’. However, the term
‘modal logic’ may be used more broadly for a family of
related systems. These include logics for belief, for tense and other
temporal expressions, for the deontic (moral) expressions such as
‘it is obligatory that’ and ‘it is permitted
that’, and many others. An understanding of modal logic is
particularly valuable in the formal analysis of philosophical argument,
where expressions from the modal family are both common and confusing.
Modal logic also has important applications in computer science.

Narrowly construed, modal logic studies reasoning that involves the use
of the expressions ‘necessarily’ and
‘possibly’. However, the term ‘modal logic’ is
used more broadly to cover a family of logics with similar rules and a
variety of different symbols.

The most familiar logics in the modal family are constructed from a
weak logic called K (after Saul Kripke). Under the narrow reading,
modal logic concerns necessity and possibility. A variety of different
systems may be developed for such logics using K as a foundation. The
symbols of K include ‘~’ for ‘not’,
‘→’ for ‘if...then’, and
‘□’ for the modal operator ‘it is necessary
that’. (The connectives ‘&’,
‘’, and ‘↔’ may be
defined from ‘~’ and ‘→’ as is done in
propositional logic.) K results from adding the following to the
principles of propositional logic.

Necessitation Rule: If A is a theorem of K,
then so is □A.

Distribution Axiom: □(A→B) →
(□A→□B).

(In these principles we use ‘A’ and ‘B’ as
metavariables ranging over formulas of the language.) According to the
Necessitation Rule, any theorem of logic is necessary. The Distribution
Axiom says that if it is necessary that if A then B, then if
necessarily A then necessarily B.

The operator ◊ (for ‘possibly’) can be defined from
□ by letting ◊A = ~□~A. In K, the operators □ and
◊ behave very much like the quantifiers ∀ (all) and ∃
(some). For example, the definition of ◊ from □ mirrors the
equivalence of ∀xA with ~∃x~A in predicate logic.
Furthermore, □(A&B) entails □A&□B and vice
versa; while
□A□B
entails
□(AB), but not vice versa.
This reflects the patterns exhibited by the universal quantifier:
∀x(A&B) entails ∀xA&∀xB and vice versa,
while
∀xA∀xB
entails
∀x(AB) but not vice versa. Similar
parallels between ◊ and ∃ can be drawn. The basis for this
correspondence between the modal operators and the quantifiers will
emerge more clearly in the section on
Possible Worlds Semantics.

The system K is too weak to provide an adequate account of
necessity. The following axiom is not provable in K, but it is clearly
desirable.

(M) □A→A

(M) claims that whatever is necessary is the case. Notice that (M)
would be incorrect were □ to be read ‘it ought to be
that’, or ‘it was the case that’. So the presence of
axiom (M) distinguishes modal from other logics in the modal family. A
basic modal logic M results from adding (M) to K. (Some authors call
this system T.)

Many logicians believe that M is still too weak to correctly
formalize the logic of necessity and possibility. They recommend
further axioms to govern the iteration, or repetition of modal
operators. Here are two of the most famous iteration axioms:

(4) □A→□□A

(5) ◊A→□◊A

S4 is the system that results from adding (4) to M. Similarly S5 is
M plus (5). In S4, the sentence □□A is equivalent to
□A. As a result, any string of boxes may be replaced by a single
box, and the same goes for strings of diamonds. This amounts to the
idea that iteration of the modal operators is superfluous. Saying that
A is necessarily necessary is considered a uselessly long-winded way of
saying that A is necessary. The system S5 has even stronger principles
for simplifying strings of modal operators. In S4, a string of
operators of the same kind can be replaced for that operator;
in S5, strings containing both boxes and diamonds are equivalent to the
last operator in the string. So, for example, saying that it is
possible that A is necessary is the same as saying that A is necessary.
A summary of these features of S4 and S5 follows.

S4: □□...□ = □
and ◊◊...◊ = ◊

S5: 00...□ = □ and 00...◊ = ◊,
where each 0 is either □ or ◊

One could engage in endless argument over the correctness or
incorrectness of these and other iteration principles for □ and
◊. The controversy can be partly resolved by recognizing that the
words ‘necessarily’ and ‘possibly’, have many
different uses. So the acceptability of axioms for modal logic depends
on which of these uses we have in mind. For this reason, there is no
one modal logic, but rather a whole family of systems built around M.
The relationship between these systems is diagrammed in
Section 8,
and their application to different uses of
‘necessarily’ and ‘possibly’ can be more deeply
understood by studying their possible world semantics in
Section 6.

The system B (for the logician Brouwer) is formed by adding axiom
(B) to M.

(B) A→□◊A

It is interesting to note that S5 can be formulated equivalently by
adding (B) to S4. The axiom (B) raises an important point about the
interpretation of modal formulas. (B) says that if A is the case, then
A is necessarily possible. One might argue that (B) should always be
adopted in any modal logic, for surely if A is the case, then it is
necessary that A is possible. However, there is a problem with this
claim that can be exposed by noting that ◊□A→A is
provable from (B). So ◊□A→A should be acceptable if (B)
is. However, ◊□A→A says that if A is possibly necessary,
then A is the case, and this is far from obvious. Why does (B) seem
obvious, while one of the things it entails seems not obvious at all?
The answer is that there is a dangerous ambiguity in the English
interpretation of A→□◊A. We often use the expression
‘If A then necessarily B’ to express that the conditional
‘if A then B’ is necessary. This interpretation corresponds
to □(A→B). On other occasions, we mean that if A, then B is
necessary: A→□B. In English, ‘necessarily’ is an
adverb, and since adverbs are usually placed near verbs, we have no
natural way to indicate whether the modal operator applies to the whole
conditional, or to its consequent. For these reasons, there is a
tendency to confuse (B): A→□◊A with
□(A→◊A). But □(A→◊A) is not the same as
(B), for □(A→◊A) is already a theorem of M, and (B) is
not. One must take special care that our positive reaction to
□(A→◊A) does not infect our evaluation of (B). One
simple way to protect ourselves is to formulate B in an equivalent way
using the axiom: ◊□A→A, where these ambiguities of scope
do not arise.

Deontic logics introduce the primitive symbol O for ‘it is
obligatory that’, from which symbols P for ‘it is permitted
that’ and F for ‘it is forbidden that’ are defined:
PA = ~O~A and FA = O~A. The deontic analog of the modal axiom (M):
OA→A is clearly not appropriate for deontic logic. (Unfortunately,
what ought to be is not always the case.) However, a basic system D of
deontic logic can be constructed by adding the weaker axiom (D) to K.

(D) OA→PA

Axiom (D) guarantees the consistency of the system of obligations by
insisting that when A is obligatory, A is permissible. A system which
obligates us to bring about A, but doesn't permit us to do so, puts us
in an inescapable bind. Although some will argue that such conflicts of
obligation are at least possible, most deontic logicians accept (D).

O(OA→A) is another deontic axiom that seems desirable. Although
it is wrong to say that if A is obligatory then A is the case
(OA→A), still, this conditional ought to be the case. So
some deontic logicians believe that D needs to be supplemented with
O(OA→A) as well.

Controversy about iteration (repetition) of operators arises again
in deontic logic. In some conceptions of obligation, OOA just amounts
to OA. ‘It ought to be that it ought to be’ is treated as a
sort of stuttering; the extra ‘ought’s do not add anything
new. So axioms are added to guarantee the equivalence of OOA and OA.
The more general iteration policy embodied in S5 may also be adopted.
However, there are conceptions of obligation where distinction between
OA and OOA is preserved. The idea is that there are genuine differences
between the obligations we actually have and the obligations
we should adopt. So, for example, ‘it ought to be that
it ought to be that A’ commands adoption of some obligation which
may not actually be in place, with the result that OOA can be true even
when OA is false.

In temporal logic (also known as tense logic), there are two basic
operators, G for the future, and H for the past. G is read ‘it
always will be that’ and the defined operator F (read ‘it
will be the case that’), can be introduced by FA = ~G~A.
Similarly H is read: ‘it always was that’ and P (for
‘it was the case that’) is defined by PA=~H~A. A basic
system of temporal logic called Kt results from adopting the principles
of K for both G and H, along with two axioms to govern the interaction
between the past and future operators:

"Necessitation" Rules: If A is a theorem then so are
GA and HA.

Distribution Axioms: G(A→B) → (GA→GB)
and H(A→B) → (HA→HB)

Interaction Axioms: A→GPA and A→HFA

The interaction axioms raise questions concerning asymmetries between
the past and the future. A standard intuition is that the past is
fixed, while the future is still open. The first interaction axiom
(A→GPA) conforms to this intuition in reporting that what is the
case (A), will at all future times, be in the past (GPA). However
A→HFA may appear to have unacceptably deterministic overtones, for
it claims, apparently, that what is true now (A) has always been such
that it will occur in the future (HFA). However, possible world
semantics for temporal logic reveals that this worry results from a
simple confusion, and that the two interaction axioms are equally
acceptable.

Note that the characteristic axiom of modal logic, (M):
□A→A, is not acceptable for either H or G, since A does not
follow from ‘it always was the case that A’, nor from
‘it always will be the case that A’. However, it is
acceptable in a closely related temporal logic where G is read
‘it is and always will be’, and H is read ‘it is and
always was’.

Depending on which assumptions one makes about the structure of
time, further axioms must be added to temporal logics. A list of axioms
commonly adopted in temporal logics follows. An account of how they
depend on the structure of time will be found in the section
Possible Worlds Semantics.

GA→GGA and HA→HHA

GGA→GA and HHA→HA

GA→FA and HA→PA

It is interesting to note that certain combinations of past tense and
futuere tense operators may be used to express complex tenses in
English. For example, FPA, corresponds to sentence A in the future
perfect tense, (as in ‘20 seconds from now the light will have
changed’). Similarly, PPA expresses the past perfect tense.

For a more detailed discussion of temporal logic, see the entry
on
temporal logic.

The founder of modal logic, C. I. Lewis, defined a series of modal
logics which did not have □ as a primitive symbol. Lewis was
concerned to develop of logic of conditionals that was free of the so
called Paradoxes of Material Implication, namely the classical theorems
A→(~A→B) and B→(A→B). He introduced the symbol
for "strict implication"
and developed logics where neither
A(~AB) nor
B(AB) is provable. The modern practice has been
to define
AB by
□(A→B), and use modal logics governing □ to obtain
similar results. However, the provability of such formulas
as
(A&~A)B in such
logics seems at odds with concern for the paradoxes. Anderson and
Belnap (1975) have developed systems R (for Relevance Logic) and E (for
Entailment) which are designed to overcome such difficulties. These
systems require revision of the standard systems of propositional
logic. (For a more detailed discussion of relevance logic, see the
entry on
relevance logic.)

David Lewis (1973) has developed special conditional logics to
handle counterfactual expressions, that is, expressions of the form
‘if A were to happen then B would
happen’. (Kvart (1980) is another good source on the topic.)
Counterfactual logics differ from those based on strict implication
because the former reject while the latter accept contraposition.

The purpose of logic is to characterize the difference between valid
and invalid arguments. A logical system for a language is a set of
axioms and rules designed to prove exactly the valid arguments
statable in the language. Creating such a logic may be a difficult
task. The logican must make sure that the system is sound,
i.e. that every argument proven using the rules and axioms is in fact
valid. Furthermore, the system should be complete, meaning
that every valid argument has a proof in the system. Demonstrating
soundness and completeness of formal systems is a logician's central
concern.

Such a demonstration cannot get underway until the concept of
validity is defined rigorously. Formal semantics for a logic provides a
definition of validity by characterizing the truth behavior of the
sentences of the system. In propositional logic, validity can be
defined using truth tables. A valid argument is simply one where every
truth table row that makes its premises true also makes its conclusion
true. However truth tables cannot be used to provide an account of
validity in modal logics because there are no truth tables for
expressions such as ‘it is necessary that’, ‘it is
obligatory that’, and the like. (The problem is that the truth
value of A does not determine the truth value for □A. For
example, when A is ‘Dogs are dogs’, □A is true, but
when A is ‘Dogs are pets’, □A is false. Nevertheless,
semantics for modal logics can be defined by introducing possible
worlds. We will illustrate possible worlds semantics for a logic of
necessity containing the symbols ~, →, and □. Then we will
explain how the same strategy may be adapted to other logics in the
modal family.

In propositional logic, a valuation of the atomic sentences (or row
of a truth table) assigns a truth value (T or F) to each propositional
variable p. Then the truth values of the complex sentences is
calculated with truth tables. In modal semantics, a set W of possible
worlds is introduced. A valuation then gives a truth value to each
propositional variable for each of the possible worlds in W.
This means that value assigned to p for world w may differ from the
value assigned to p for another world w′.

The truth value of the atomic sentence p at world w given by the
valuation v may be written v(p, w). Given this notation, the truth
values (T for true, F for false) of complex sentences of modal logic
for a given valuation v (and member w of the set of worlds W) may be
defined by the following truth clauses. (‘iff’ abbreviates
‘if and only if’.)

(~) v(~A, w)=T iff v(A,
w)=F.

(→) v(A→B, w)=T iff v(A, w)=F or v(B,
w)=T.

(5) v(□A, w)=T iff for every
world w′ in W, v(A, w′)=T.

Clauses (~) and (→) simply describe the standard truth table
behavior for negation and material implication respectively. According
to (5), □A is true (at a world w) exactly when A is true in
all possible worlds. Given the definition of ◊, (namely,
◊A = ~□~A) the truth condition (5) insures that ◊A is
true just in case A is true in some possible world. Since the
truth clauses for □ and ◊ involve the quantifiers
‘all’ and ‘some’ (respectively), the parallels
in logical behavior between □ and ∀x, and between ◊
and ∃x noted in section 2 will be expected.

Clauses (~), (→), and (5) allow us to calculate the truth value
of any sentence at any world on a given valuation. A definition of
validity is now just around the corner. An argument is 5-valid for
a given set W (of possible worlds) if and only if every valuation
of the atomic sentences that assigns the premises T at a world in W
also assigns the conclusion T at the same world. An argument is said to
be 5-valid iff it is valid for every non empty set of W of
possible worlds.

It has been shown that S5 is sound and complete for 5-validity
(hence our use of the symbol ‘5’). The 5-valid arguments
are exactly the arguments provable in S5. This result suggests that S5
is the correct way to formulate a logic of necessity.

However, S5 is not a reasonable logic for all members of the modal
family. In deontic logic, temporal logic, and others, the analog of the
truth condition (5) is clearly not appropriate; furthermore there are
even conceptions of necessity where (5) should be rejected as well. The
point is easiest to see in the case of temporal logic. Here, the
members of W are moments of time, or worlds "frozen", as it were, at an
instant. For simplicity let us consider a future temporal
logic, a logic where □A reads: ‘it will always be
the case that’. (We formulate the system using □ rather
than the traditional G so that the connections with other modal logics
will be easier to appreciate.) The correct clause for □ should
say that □A is true at time w iff A is true at all times in
the future of w. To restrict attention to the future, the relation
R (for ‘eaRlier than’) needs to be introduced. Then the
correct clause can be formulated as follows.

(K) v(□A, w)=T iff for every
w′, if wRw′, then v(A, w′)=T.

This says that □A is true at w just in case A is true at all
times after w.

Validity for this brand of temporal logic can now be defined. A
frame <W, R> is a pair consisting of a non-empty set W
(of worlds) and a binary relation R on W. A model <F, v>
consists of a frame F, and a valuation v that assigns truth values to
each atomic sentence at each world in W. Given a model, the values of
all complex sentences can be determined using (~), (→), and (K).
An argument is K-valid just in case any model whose valuation assigns
the premises T at a world also assigns the conclusion T at the same
world. As the reader may have guessed from our use of ‘K’,
it has been shown that the simplest modal logic K is both sound and
complete for K-validity.

One might assume from this discussion that K is the correct logic when
□ is read ‘it will always be the case that’. However,
there are reasons for thinking that K is too weak. One obvious logical
feature of the relation R (earlier than) is transitivity. If wRv (w is
earlier than v) and vRu (v is earlier than u), then it follows that wRu
(w is earlier than u). So let us define a new kind of validity that
corresponds to this condition on R. Let a 4-model be any model whose
frame <W, R> is such that R is a transitive relation on W. Then
an argument is 4-valid iff any 4-model whose valuation assigns T to the
premises at a world also assigns T to the conclusion at the same world.
We use ‘4’ to describe such a transitive model because the
logic which is adequate (both sound and complete) for 4-validity is K4,
the logic which results from adding the axiom (4):
□A→□□A to K.

Transitivity is not the only property which we might want to require
of the frame <W, R> if R is to be read ‘earlier than’
and W is a set of moments. One condition (which is only mildly
controversial) is that there is no last moment of time, i.e. that for
every world w there is some world v such that wRv. This condition on
frames is called seriality. Seriality corresponds to the axiom
(D): □A→◊A, in the same way that transitivity
corresponds to (4). A D-model is a K-model with a serial frame. From
the concept of a D-model the corresponding notion of D-validity can be
defined just as we did in the case of 4-validity. As you probably
guessed, the system that is adequate with respect to D-validity is KD,
or K plus (D). Not only that, but the system KD4 (that is K plus (4)
and (D)) is adequate with respect to D4-validity, where a D4-model is
one where <W, R> is both serial and transitive.

Another property which we might want for the relation ‘earlier
than’ is density, the condition which says that between any two
times we can always find another. Density would be false if time were
atomic, i.e. if there were intervals of time which could not be broken
down into any smaller parts. Density corresponds to the axiom (C4):
□□A→□A, the converse of (4), so for example, the
system KC4, which is K plus (C4) is adequate with respect to models
where the frame <W, R> is dense, and KDC4, adequate with respect
to models whose frames are serial and dense, and so on.

Each of the modal logic axioms we have discussed corresponds to a
condition on frames in the same way. The relationship between
conditions on frames and corresponding axioms is one of the central
topics in the study of modal logics. Once an interpretation of the
intensional operator □ has been decided on, the appropriate
conditions on R can be determined to fix the corresponding notion of
validity. This, in turn, allows us to select the right set of axioms
for that logic.

For example, consider a deontic logic, where □ is read
‘it is obligatory that’. Here the truth of □A does
not demand the truth of A in every possible world, but only in
a subset of those worlds where people do what they ought. So we will
want to introduce a relation R for for this kind of logic as well, and
use the truth clause (K) to evaluate □A at a world. However, in
this case, R is not earlier than. Instead wRw′ holds just in case
world w′ is a morally acceptable variant of w, i.e. a world that
our actions can bring about which satisfies what is morally correct, or
right, or just. Under such a reading, it should be clear that the
relevant frames should obey seriality, the condition that requires that
each possible world have a morally acceptable variant. The analysis of
the properties desired for R makes it clear that a basic deontic logic
can be formulated by adding the axiom (D) and to K.

Even in modal logic, one may wish to restrict the range of possible
worlds which are relevant in determining whether □A is true at a
given world. For example, I might say that it is necessary for me to
pay my bills, even though I know full well that there is a possible
world where I fail to pay them. In ordinary speech, the claim that A is
necessary does not require the truth of A in all possible
worlds, but rather only in a certain class of worlds which I have in
mind (for example, worlds where I avoid penalties for failure to pay).
In order to provide a generic treatment of necessity, we must say that
□A is true in w iff A is true in all worlds that are
related to w in the right way. So for an operator □
interpreted as necessity, we introduce a corresponding relation R on
the set of possible worlds W, traditionally called the accessibility
relation. The accessibility relation R holds between worlds w and
w′ iff w′ is possible given the facts of w. Under this
reading for R, it should be clear that frames for modal logic should be
reflexive. It follows that modal logics should be founded on M, the
system that results from adding (M) to K. Depending on exactly how the
accessibility relation is understood, symmetry and transitivity may
also be desired.

A list of some of the more commonly discussed conditions on frames
and their corresponding axioms along with a map showing the
relationship between the various modal logics can be found in the next
section.

The following diagram shows the relationships between the best known
modal logics, namely logics that can be formed by adding a selection of
the axioms (D), (M), (4), (B) and (5) to K. A list of these (and other)
axioms along with their corresponding frame conditions can be found
below the diagram.

In this chart, systems are given by the list of their axioms. So, for
example M4B is the result of adding (M) (4) and (B) to K. In boldface,
we have indicated traditional names of some systems. When system S
appears below and/or to the left of S′ connected by a line, then
S′ is an extension of S. This means that every argument provable
in S is provable in S′, but S is weaker than S′, i.e. not
all arguments provable in S′ are provable in S.

The following list indicates axioms, their names, and the
corresponding conditions on the accessibility relation R, for axioms so
far discussed in this encyclopedia entry.

Axiom
Name

Axiom

Condition on Frames

R is...

(D)

□A→◊A

∃u wRu

Serial

(M)

□A→A

wRw

Reflexive

(4)

□A→□□A

(wRv&vRu) ⇒ wRu

Transitive

(B)

A→□◊A

wRv ⇒ vRw

Symmetric

(5)

◊A→□◊A

(wRv&wRu) ⇒ vRu

Euclidean

(CD)

◊A→□A

(wRv&wRu) ⇒ v=u

Unique

(□M)

□(□A→A)

wRv ⇒ vRv

Shift Reflexive

(C4)

□□A→□A

wRv ⇒ ∃u(wRu&uRv)

Dense

(C)

◊□A → □◊A

wRv&wRx ⇒ ∃u(vRu&xRu)

Convergent

In the list of conditions on frames, the variables ‘w’,
‘v’, ‘u’, ‘x’ and the quantifier
‘∃u’ are understood to range over W.
‘&’ abbreviates ‘and’ and
‘⇒’ abbreviates ‘if...then’.

The correspondence between axioms and conditions on frames may seem
something of a mystery. A beautiful result of Lemmon and Scott (1977)
goes a long way towards explaining those relationships. Their theorem
concerned axioms which have the following form:

(G) ◊h□iA →
□j◊kA

We use the notation ‘◊n’ to represent n
diamonds in a row, so, for example, ‘◊3’
abbreviates a string of three diamonds: ‘◊◊◊’.
Similarly ‘□n’ represents a string of n
boxes. When the values of h, i, j, and k are all 1, we have axiom (C):

(C) ◊□A → □◊A =
◊1□1A →
□1◊1A

The axiom (B) results from setting h and k to 0, and letting j and k be
1:

(B) A → □◊A =
◊0□0A →
□1◊1A

To obtain (4), we may set h and k to 0, set i to 1 and j to 2:

(4) □A →□□A =
◊0□1A →
□2◊0A

Many (but not all) axioms of modal logic can be obtained by setting the
right values for the parameters in (G)

Our next task will be to give the condition on frames which
corresponds to (G) for a given selection of values for h, i, j, and k.
In order to do so, we will need a definition. The composition of two
relations R and R′ is a new relation
RR′ which is defined as follows:

wRR′v
iff for some u, wRu and uR′v.

For example, if R is the relation of being a brother, and R′ is
the relation of being a parent then
RR′ is the relation of being an uncle,
(because w is the uncle of v iff for some person u, both w is the
brother of u and u is the parent of v). A relation may be composed with
itself. For example, when R is the relation of being a parent,
then
RR is the relation of being a
grandparent, and
RRR is the relation of being a
great-grandparent. It will be useful to write
‘Rn’, for the result of composing R with itself
n times. So R2 is
RR, and R4 is
RRRR. We will let R1 be R,
and R0 will be the identity relation, i.e. wR0v
iff w=v.

We may now state the Scott-Lemmon result. It is that the condition
on frames which corresponds exactly to any axiom of the shape (G) is
the following.

(hijk-Convergence) wRhv &
wRju ⇒ ∃x (vRix &
uRkx)

It is interesting to see how the familiar conditions on R result from
setting the values for h, i, j, and k according to the values in the
corresponding axiom. For example, consider (5). In this case i=0, and
h=j=k=1. So the corresponding condition is

wRv & wRu ⇒ ∃x (vR0x &
uRx).

We have explained that R0 is the identity relation. So if
vR0x then v=x. But ∃x (v=x & uRx), is equivalent
to uRv, and so the Euclidean condition is obtained:

(wRv & wRu) ⇒ uRv.

In the case of axiom (4), h=0, i=1, j=2 and k=0. So the corresponding
condition on frames is

(w=v & wR2u) ⇒ ∃x (vRx &
u=x).

Resolving the identities this amounts to:

vR2u ⇒ vRu.

By the definition of R2, vR2u iff ∃x(vRx
& xRu), so this comes to:

∃x(vRx & xRu) ⇒ vRu,

which by predicate logic, is equivalent to transitivity.

vRx & xRu ⇒ vRu.

The reader may find it a pleasant exercise to see how the corresponding
conditions fall out of hijk-Convergence when the values of the
parameters h, i, j, and k are set by other axioms.

The Scott-Lemmon results provides a quick method for establishing
results about the relationship between axioms and their corresponding
frame conditions. Since they showed the adequacy of any logic that
extends K with a selection of axioms of the form (G) with respect to
models that satisfy the corresponding set of frame conditions, they
provided "wholesale" adequacy proofs for the majority of systems in the
modal family. Sahlqvist (1975) has discovered important generalizations
of the Scott-Lemmon result covering a much wider range of axiom
types.

Modal logic has been useful in clarifying our understanding of central
results concerning provability in the foundations of mathematics
(Boolos, 1993). Provability logics are systems where the propositional
variables p, q, r, etc. range over formulas of some mathematical
system, for example Peano's system PA for arithmetic. (The system
chosen for mathematics might vary, but assume it is PA for this
discussion.) Gödel showed that arithmetic has strong expressive
powers. Using code numbers for arithmetic sentences, he was able to
demonstrate a correspondence between sentences of mathematics and facts
about which sentences are and are not provable in PA. For example, he
showed there there is a sentence C that is true just in case no
contradiction is provable in PA and there is a sentence G (the famous
Gödel sentence) that is true just in case it is not provable in
PA.

In provability logics, □p is interpreted as a formula (of
arithmetic) that expresses that what p denotes is provable in PA. Using
this notation, sentences of provability logic express facts about
provability. Suppose that
is a constant of provability logic denoting a
contradiction. Then
~□ says that PA is consistent and □A→A says
that PA is sound in the sense that when it proves A, A is indeed true.
Furthermore, the box may be iterated. So, for
example,
□~□
makes the dubious claim that PA is able to prove its own consistency,
and
~□
→
~□~□ asserts (correctly as Gödel proved) that if PA
is consistent then PA is unable to prove its own consistency.

Although provability logics form a family of related systems, the
system GL is by far the best known. It results from adding the
following axiom to K:

(GL)
□(□A→A)→□A

The axiom (4): □A→□□A is provable in GL, so GL
is actually a strengthening of K4. However, axioms such as (M):
□A→A, and even the weaker (D): □A→◊A are not
available (nor desirable) in GL. In provability logic, provability is
not to be treated as a brand of necessity. The reason is that when p is
provable in an arbitrary system S for mathematics, it does not follow
that p is true, since S may be unsound. Furthermore, if p is provable
in S (□p) it need not even follow that ~p lacks a proof
(~□~p = ◊p). S might be inconsistent and so prove both p and
~p.

Axiom (GL) captures the content of Loeb's Theorem, an important
result in the foundations of arithmetic. □A→A says that PA
is sound for A, i.e. that if A were proven, A would be true. (Such a
claim might not be secure for an arbitrarily selected sytem S, since A
might be provable in S and false.) (GL) claims that if PA manages to
prove the sentence that claims soundness for a given sentence A, then A
is already provable in PA. Loeb's Theorem reports a kind of modesty on
PA's part (Boolos, 1993, p. 55). PA never insists (proves) that a proof
of A entails A's truth, unless it already has a proof of A to back up
that claim.

It has been shown that GL is adequate for provability in the
following sense. Let a sentence of GL be always provable
exactly when the sentence of arithmetic it denotes is provable no
matter how its variables are assigned values to sentences of PA. Then
the provable sentences of GL are exactly the sentences that are always
provable. This adequacy result has been extremely useful, since general
questions concerning provability in PA can be transformed into easier
questions about what can be demonstrated in GL.

GL can also be outfitted with a possible world semantics for which
it is sound and complete. A corresponding condition on frames for
GL-validity is that the frame be transitive, finite and
irreflexive.

It would seem to be a simple matter to outfit a modal logic with the
quantifiers ∀ (all) and ∃ (some). One would simply add the
standard (or classical) rules for quantifiers to the principles of
whichever propositional modal logic one chooses. However, systems of
this kind create problems which have motivated some logicians to
abandon classical quantifier rules in favor of the weaker rules of free
logic (Garson, 1984). The controversy over whether classical principles
should be adopted continues today.

The main points of disagreement can be traced back to decisions
about how to handle the domain of quantification. The simplest
alternative, the fixed-domain (sometimes called the possibilist)
approach, assumes a single domain of quantification that contains all
the possible objects. On the other hand, the world-relative (or
actualist) interpretation, assumes that the domain of quantification
changes from world to world, and contains only the objects that
actually exist in a given world.

The fixed-domain approach requires no major adjustments to the
classical machinery for the quantifiers. Modal logics that are adequate
for fixed domain semantics can usually be axiomatized by adding
principles of a propositional modal logic to classical quantifier rules
together with the Barcan Formula (BF) (Barcan 1946). (For an account of
some interesting exceptions see Cresswell (1995)).

(BF)
∀x□A→□∀xA.

The fixed-domain interpretation has advantages of simplicity and
familiarity, but it does not provide a direct account of the semantics
of certain quantifier expressions of natural language. We do not think
that ‘Some man exists who signed the Declaration of
Independence’ is true, at least not if we read
‘exists’ in the present tense. Nevertheless, this sentence
was true in 1777, which shows that the domain for the natural language
expression ‘some man exists who’ changes to reflect which
men exist at different times. A related problem is that on the
fixed-domain interpretation, the sentence ∀y□∃x(x=y)
is valid. However, assuming that ∃x(x=y) is read: y exists,
∀y□∃x(x=y) says that everything exists necessarily.
However, it seems a fundamental feature of common ideas about modality
that the existence of many things is contingent, and that different
objects exist in different possible worlds.

The defender of the fixed-domain interpretation may respond to these
objections by insisting that on his (her) reading of the quantifiers,
the domain of quantification contains all possible objects,
not just the objects that happen to exist at a given world. So the
theorem ∀y□∃x(x=y) makes the innocuous claim that
every possible object is necessarily found in the domain of
all possible objects. Furthermore, those quantifier expressions of
natural language whose domain is world (or time) dependent can be
expressed using the fixed-domain quantifier ∃x and a predicate
letter E with the reading ‘actually exists’. For example,
instead of translating ‘Some Man exists who
Signed the Declaration of Independence’ by

∃x(Mx&Sx),

the defender of fixed domains may write:

∃x(Ex&Mx&Sx),

thus ensuring the translation is counted false at the present time.
Cresswell (1991) makes the interesting observation that world-relative
quantification has limited expressive power relative to fixed-domain
quantification. World-relative quantification can be defined with fixed
domain quantifiers and E, but there is no way to fully express
fixed-domain quantifiers with world-relative ones. Although this argues
in favor of the classical approach to quantified modal logic, the
translation tactic also amounts to something of a concession in favor
of free logic, for the world-relative quantifiers so defined obey
exactly the free logic rules.

A problem with the translation strategy used by defenders of fixed
domain quantification is that rendering the English into logic is less
direct, since E must be added to all translations of all sentences
whose quantifier expressions have domains that are context dependent. A
more serious objection to fixed-domain quantification is that it strips
the quantifier of a role which Quine recommended for it, namely to
record robust ontological commitment. On this view, the domain of
∃x must contain only entities that are ontologically respectable,
and possible objects are too abstract to qualify. Actualists of this
stripe will want to develop the logic of a quantifier ∃x which
reflects commitment to what is actual in a given world rather than to
what is merely possible.

However, recent work on
actualism
tends
to undermine this objection. For example, Linksy and Zalta (1994) argue
that the fixed-domain quantifier can be given an interpretation that is
perfectly acceptable to actualists. Actualists who employ possible
worlds semantics routinely quantify over possible worlds in their
semantical theory of language. So it would seem that possible worlds
are actual by these actualist's lights. By cleverly outfitting the
domain with abstract entities no more objectionable than the ones
actualists accept, Linsky and Zalta show that the Barcan Formula and
classical principles can be vindicated. Note however, that actualists
may respond that they need not be commited to the actuality of possible
worlds so long as it is understood that quantifiers used in their
theory of language lack strong ontological import. In any case, it is
open to actualists (and non actualists as well) to investigate the
logic of quantifiers with more robust domains, for example domains
excluding possible worlds and other such abstract entities, and
containing only the spatio-temporal particulars found in a given world.
For quantifiers of this kind, a world-relative domains are
appropriate.

Such considerations motivate interest in systems that acknowledge
the context dependence of quantification by introducing world-relative
domains. Here each possible world has its own domain of quantification
(the set of objects that actually exist in that world), and the domains
vary from one world to the next. When this decision is made, a
difficulty arises for classical quantification theory. Notice that the
sentence ∃x(x=t) is a theorem of classical logic, and so
□∃x(x=t) is a theorem of K by the Necessitation Rule. Let
the term t stand for Saul Kripke. Then this theorem says that it is
necessary that Saul Kripke exists, so that he is in the domain of every
possible world. The whole motivation for the world-relative approach
was to reflect the idea that objects in one world may fail to exist in
another. If standard quantifier rulers are used, however, every term t
must refer to something that exists in all the possible worlds. This
seems incompatible with our ordinary practice of using terms to refer
to things that only exist contingently.

One response to this difficulty is simply to eliminate terms. Kripke
(1963) gives an example of a system that uses the world-relative
interpretation and preserves the classical rules. However, the costs
are severe. First, his language is artificially impoverished, and
second, the rules for the propositional modal logic must be
weakened.

Presuming that we would like a language that includes terms, and
that classical rules are to be added to standard systems of
propositional modal logic, a new problem arises. In such a system, it
is possible to prove (CBF), the converse of the Barcan Formula.

(CBF)
□∀xA→∀x□A.

This fact has serious consequences for the system's semantics. It is
not difficult to show that every world-relative model of (CBF) must
meet condition (ND) (for ‘nested domains’).

(ND) If wRv then the domain of w is a subset of the
domain of v.

However (ND) conflicts with the point of introducing world-relative
domains. The whole idea was that existence of objects is contingent so
that there are accessible possible worlds where one of the things in
our world fails to exist.

A straightforward solution to these problems is to abandon classical
rules for the quantifiers and to adopt rules for free logic (FL)
instead. The rules of FL are the same as the classical rules, except
that inferences from ∀xRx (everything is real) to Rp (Pegasus is
real) are blocked. This is done by introducing a predicate
‘E’ (for ‘actually exists’) and modifying the
rule of universal instantiation. From ∀xRx one is allowed to
obtain Rp only if one also has obtained Ep. Assuming that the universal
quantifier ∀x is primitive, and the existential quantifier
∃x is defined by ∃xA =df ~∀x~A, then FL may be
constructed by adding the following two principles to the rules of
propositional logic

Universal Generalization. If B→A(y) is a
theorem, so is B→∀xA(x).

Universal Instantiation. (∀xA(x) &
En)→A(n)

(Here it is assumed that A(x) is any well-formed formula of predicate
logic, and that A(y) and A(n) result from replacing y and n properly
for each occurrence of x in A(x).) Note that the principle of universal
generalization is standard, but that the instantiation axiom is
restricted by mention of En in the antecedent. In FL, proofs of
formulas like ∃x□(x=t), ∀y□ ∃x(x=y),
(CBF), and (BF) which seem incompatible with the world-relative
interpretation, are blocked.

One philosophical objection to FL is that E appears to be an
existence predicate, and many would argue that existence is not a
legitimate property like being green or weighing more than four pounds.
So philosophers who reject the idea that existence is a predicate may
object to FL. However in most (but not all) quantified modal logics
that include identity (=) these worries may be skirted by defining E as
follows.

Et =df ∃x(x=t).

The most general way to formulate quantified modal logic is to create
FS by adding the rules of FL to a given propositional modal logic S. In
situations where classical quantification is desired, one may simply
add Et as an axiom to FS, so that the classical principles become
derivable rules. Adequacy results for such systems can be obtained for
most choices of the modal logic S, but there are exceptions.

A final complication in the semantics for quantified modal logic is
worth mentioning. It arises when non-rigid expressions such as
‘the inventor of bifocals’, are introduced to the language.
A term is non-rigid when it picks out different objects in different
possible worlds. The semantical value of such a term can be given by
what Carnap (1947) called an individual concept, a function that picks
out the denotation of the term for each possible world. One approach to
dealing with non-rigid terms is to employ Russell's theory of
descriptions. However, in a language that treats non rigid expressions
as genuine terms, it turns out that neither the classical nor the free
logic rules for the quantifiers are acceptable. (The problem can not be
resolved by weakening the rule of substitution for identity.) A
solution to this problem is to employ a more general treatment of the
quantifiers, where the domain of quantification contains individual
concepts rather than objects. This more general interpretation provides
a better match between the treatment of terms and the treatment of
quantifiers and results in systems that are adequate for classical or
free logic rules (depending on whether the fixed domains or
world-relative domains are chosen).

Sahlqvist, H., 1975, "Completeness and Correspondence in First and
Second Order Semantics for Modal Logic," in Kanger, S. (ed.)
Proceedings of the Third Scandanavian Logic Symposium,
Amsterdam: North Holland. 110-143